manu/mldr-zoomed-100char-total-queries-documents

chunk_id stringlengths 3 9	chunk stringlengths 1 100
4_0	Bridgewater is a town in Aroostook County, Maine, United States. The population was 532 at the 2020
4_1	census.
4_2	Geography
4_3	According to the United States Census Bureau, the town has a total area of , of which is land and
4_4	is water.
4_5	Climate
4_6	This climatic region is typified by large seasonal temperature differences, with warm to hot (and
4_7	often humid) summers and cold (sometimes severely cold) winters. According to the Köppen Climate
4_8	Classification system, Bridgewater has a humid continental climate, abbreviated "Dfb" on climate
4_9	maps.
4_10	Demographics
4_11	2010 census
4_12	As of the census of 2010, there were 610 people, 263 households, and 175 families living in the
4_13	town. The population density was . There were 326 housing units at an average density of . The
4_14	racial makeup of the town was 96.7% White, 0.7% Native American, 0.2% Asian, 1.0% from other races,
4_15	and 1.5% from two or more races. Hispanic or Latino of any race were 1.1% of the population.
4_16	There were 263 households, of which 25.9% had children under the age of 18 living with them, 55.5%
4_17	were married couples living together, 8.4% had a female householder with no husband present, 2.7%
4_18	had a male householder with no wife present, and 33.5% were non-families. 29.7% of all households
4_19	were made up of individuals, and 14.4% had someone living alone who was 65 years of age or older.
4_20	The average household size was 2.32 and the average family size was 2.90.
4_21	The median age in the town was 46.7 years. 22.5% of residents were under the age of 18; 4.4% were
4_22	between the ages of 18 and 24; 20.7% were from 25 to 44; 32.7% were from 45 to 64; and 19.8% were
4_23	65 years of age or older. The gender makeup of the town was 50.0% male and 50.0% female.
4_24	2000 census
4_25	As of the census of 2000, there were 612 people, 248 households, and 173 families living in the
4_26	town. The population density was 15.8 people per square mile (6.1/km2). There were 316 housing
4_27	units at an average density of 8.1 per square mile (3.1/km2). The racial makeup of the town was
4_28	98.04% White, 0.49% Native American, 0.65% from other races, and 0.82% from two or more races.
4_29	Hispanic or Latino of any race were 0.65% of the population.
4_30	There were 248 households, out of which 27.0% had children under the age of 18 living with them,
4_31	59.3% were married couples living together, 6.0% had a female householder with no husband present,
4_32	and 30.2% were non-families. 25.4% of all households were made up of individuals, and 12.9% had
4_33	someone living alone who was 65 years of age or older. The average household size was 2.47 and the
4_34	average family size was 2.97.
4_35	In the town, the population was spread out, with 22.2% under the age of 18, 5.6% from 18 to 24,
4_36	25.0% from 25 to 44, 26.6% from 45 to 64, and 20.6% who were 65 years of age or older. The median
4_37	age was 43 years. For every 100 females, there were 101.3 males. For every 100 females age 18 and
4_38	over, there were 94.3 males.
4_39	The median income for a household in the town was $27,679, and the median income for a family was
4_40	$33,125. Males had a median income of $24,167 versus $21,190 for females. The per capita income for
4_41	the town was $15,534. About 12.7% of families and 17.6% of the population were below the poverty
4_42	line, including 23.7% of those under age 18 and 16.5% of those age 65 or over.
4_43	History and settlement
4_44	In 1820 the State of Maine was officially separated from Massachusetts, and at that time the name
4_45	Bridgewater was applied to the Township. The area north of Bangor had been previously divided into
4_46	6 mile square townships, and in 1803 the future Bridgewater Township was subdivided into two 3 mile
4_47	x 6 mile areas, each designated a "grant" area to fund public academies in Portland and
4_48	Bridgewater, respectively. The town of Bridgewater was incorporated on 2 March 1858.
4_49	Notable people
4_50	Jim Gerritsen, organic potato farmer and anti-GMO activist
4_51	Colonel Frank M. Hume, commanding officer of the 103rd Infantry, 26th Division during World War I
4_52	Colonel Gerald Evan Williams, World War II Air Force officer
4_53	Sites of interest Bridgewater Town Hall and Jail References External links
4_54	Towns in Aroostook County, Maine Towns in Maine
5_0	In physics, the Fermi–Pasta–Ulam–Tsingou problem or formerly the Fermi–Pasta–Ulam problem was the
5_1	apparent paradox in chaos theory that many complicated enough physical systems exhibited almost
5_2	exactly periodic behavior – called Fermi–Pasta–Ulam–Tsingou recurrence (or Fermi–Pasta–Ulam
5_3	recurrence) – instead of the expected ergodic behavior. This came as a surprise, as Fermi,
5_4	certainly, expected the system to thermalize in a fairly short time. That is, it was expected for
5_5	all vibrational modes to eventually appear with equal strength, as per the equipartition theorem,
5_6	or, more generally, the ergodic hypothesis. Yet here was a system that appeared to evade the
5_7	ergodic hypothesis. Although the recurrence is easily observed, it eventually became apparent that
5_8	over much, much longer time periods, the system does eventually thermalize. Multiple competing
5_9	theories have been proposed to explain the behavior of the system, and it remains a topic of active
5_10	research.
5_11	The original intent was to find a physics problem worthy of numerical simulation on the then-new
5_12	MANIAC computer. Fermi felt that thermalization would pose such a challenge. As such, it represents
5_13	one of the earliest uses of digital computers in mathematical research; simultaneously, the
5_14	unexpected results launched the study of nonlinear systems.
5_15	The FPUT experiment
5_16	In the summer of 1953 Enrico Fermi, John Pasta, Stanislaw Ulam, and Mary Tsingou conducted computer
5_17	simulations of a vibrating string that included a non-linear term (quadratic in one test, cubic in
5_18	another, and a piecewise linear approximation to a cubic in a third). They found that the behavior
5_19	of the system was quite different from what intuition would have led them to expect. Fermi thought
5_20	that after many iterations, the system would exhibit thermalization, an ergodic behavior in which
5_21	the influence of the initial modes of vibration fade and the system becomes more or less random
5_22	with all modes excited more or less equally. Instead, the system exhibited a very complicated
5_23	quasi-periodic behavior. They published their results in a Los Alamos technical report in 1955.
5_24	(Enrico Fermi died in 1954, and so this technical report was published after Fermi's death.)
5_25	In 2020, National Security Science magazine featured an article on Tsingou that included her
5_26	commentary and historical reflections on the FPUT problem. In the article,Tsingou states “I
5_27	remember sitting there one day with Pasta and Ulam,” as they brainstormed “some problems we could
5_28	do on the computer, some really mathematical problems.” They tried several things, but, eventually,
5_29	“they came up with this vibrating string.”
5_30	The FPUT experiment was important both in showing the complexity of nonlinear system behavior and
5_31	the value of computer simulation in analyzing systems.
5_32	Name change
5_33	The original paper names Fermi, Pasta, and Ulam as authors (although Fermi died before the report
5_34	was written) with an acknowledgement to Tsingou for her work in programming the MANIAC simulations.
5_35	Mary Tsingou's contributions to the FPUT problem were largely ignored by the community until
5_36	published additional information regarding the development and called for the problem to be renamed
5_37	to grant her attribution as well.
5_38	The FPUT lattice system
5_39	Fermi, Pasta, Ulam, and Tsingou simulated the vibrating string by solving the following discrete
5_40	system of nearest-neighbor coupled oscillators. We follow the explanation as given in Richard
5_41	Palais's article. Let there be N oscillators representing a string of length with equilibrium
5_42	positions , where is the lattice spacing. Then the position of the j-th oscillator as a function
5_43	of time is , so that gives the displacement from equilibrium. FPUT used the following equations of
5_44	motion:

4_0

Bridgewater is a town in Aroostook County, Maine, United States. The population was 532 at the 2020

4_1

census.

4_2

Geography

4_3

According to the United States Census Bureau, the town has a total area of , of which is land and

4_4

is water.

4_5

Climate

4_6

This climatic region is typified by large seasonal temperature differences, with warm to hot (and

4_7

often humid) summers and cold (sometimes severely cold) winters. According to the Köppen Climate

4_8

Classification system, Bridgewater has a humid continental climate, abbreviated "Dfb" on climate

4_9

maps.

4_10

Demographics

4_11

2010 census

4_12

As of the census of 2010, there were 610 people, 263 households, and 175 families living in the

4_13

town. The population density was . There were 326 housing units at an average density of . The

4_14

racial makeup of the town was 96.7% White, 0.7% Native American, 0.2% Asian, 1.0% from other races,

4_15

and 1.5% from two or more races. Hispanic or Latino of any race were 1.1% of the population.

4_16

There were 263 households, of which 25.9% had children under the age of 18 living with them, 55.5%

4_17

were married couples living together, 8.4% had a female householder with no husband present, 2.7%

4_18

had a male householder with no wife present, and 33.5% were non-families. 29.7% of all households

4_19

were made up of individuals, and 14.4% had someone living alone who was 65 years of age or older.

4_20

The average household size was 2.32 and the average family size was 2.90.

4_21

The median age in the town was 46.7 years. 22.5% of residents were under the age of 18; 4.4% were

4_22

between the ages of 18 and 24; 20.7% were from 25 to 44; 32.7% were from 45 to 64; and 19.8% were

4_23

65 years of age or older. The gender makeup of the town was 50.0% male and 50.0% female.

4_24

2000 census

4_25

As of the census of 2000, there were 612 people, 248 households, and 173 families living in the

4_26

town. The population density was 15.8 people per square mile (6.1/km2). There were 316 housing

4_27

units at an average density of 8.1 per square mile (3.1/km2). The racial makeup of the town was

4_28

98.04% White, 0.49% Native American, 0.65% from other races, and 0.82% from two or more races.

4_29

Hispanic or Latino of any race were 0.65% of the population.

4_30

There were 248 households, out of which 27.0% had children under the age of 18 living with them,

4_31

59.3% were married couples living together, 6.0% had a female householder with no husband present,

4_32

and 30.2% were non-families. 25.4% of all households were made up of individuals, and 12.9% had

4_33

someone living alone who was 65 years of age or older. The average household size was 2.47 and the

4_34

average family size was 2.97.

4_35

In the town, the population was spread out, with 22.2% under the age of 18, 5.6% from 18 to 24,

4_36

25.0% from 25 to 44, 26.6% from 45 to 64, and 20.6% who were 65 years of age or older. The median

4_37

age was 43 years. For every 100 females, there were 101.3 males. For every 100 females age 18 and

4_38

over, there were 94.3 males.

4_39

The median income for a household in the town was $27,679, and the median income for a family was

4_40

$33,125. Males had a median income of $24,167 versus $21,190 for females. The per capita income for

4_41

the town was $15,534. About 12.7% of families and 17.6% of the population were below the poverty

4_42

line, including 23.7% of those under age 18 and 16.5% of those age 65 or over.

4_43

History and settlement

4_44

In 1820 the State of Maine was officially separated from Massachusetts, and at that time the name

4_45

Bridgewater was applied to the Township. The area north of Bangor had been previously divided into

4_46

6 mile square townships, and in 1803 the future Bridgewater Township was subdivided into two 3 mile

4_47

x 6 mile areas, each designated a "grant" area to fund public academies in Portland and

4_48

Bridgewater, respectively. The town of Bridgewater was incorporated on 2 March 1858.

4_49

Notable people

4_50

Jim Gerritsen, organic potato farmer and anti-GMO activist

4_51

Colonel Frank M. Hume, commanding officer of the 103rd Infantry, 26th Division during World War I

4_52

Colonel Gerald Evan Williams, World War II Air Force officer

4_53

Sites of interest Bridgewater Town Hall and Jail References External links

4_54

Towns in Aroostook County, Maine Towns in Maine

5_0

In physics, the Fermi–Pasta–Ulam–Tsingou problem or formerly the Fermi–Pasta–Ulam problem was the

5_1

apparent paradox in chaos theory that many complicated enough physical systems exhibited almost

5_2

exactly periodic behavior – called Fermi–Pasta–Ulam–Tsingou recurrence (or Fermi–Pasta–Ulam

5_3

recurrence) – instead of the expected ergodic behavior. This came as a surprise, as Fermi,

5_4

certainly, expected the system to thermalize in a fairly short time. That is, it was expected for

5_5

all vibrational modes to eventually appear with equal strength, as per the equipartition theorem,

5_6

or, more generally, the ergodic hypothesis. Yet here was a system that appeared to evade the

5_7

ergodic hypothesis. Although the recurrence is easily observed, it eventually became apparent that

5_8

over much, much longer time periods, the system does eventually thermalize. Multiple competing

5_9

theories have been proposed to explain the behavior of the system, and it remains a topic of active

5_10

research.

5_11

The original intent was to find a physics problem worthy of numerical simulation on the then-new

5_12

MANIAC computer. Fermi felt that thermalization would pose such a challenge. As such, it represents

5_13

one of the earliest uses of digital computers in mathematical research; simultaneously, the

5_14

unexpected results launched the study of nonlinear systems.

5_15

The FPUT experiment

5_16

In the summer of 1953 Enrico Fermi, John Pasta, Stanislaw Ulam, and Mary Tsingou conducted computer

5_17

simulations of a vibrating string that included a non-linear term (quadratic in one test, cubic in

5_18

another, and a piecewise linear approximation to a cubic in a third). They found that the behavior

5_19

of the system was quite different from what intuition would have led them to expect. Fermi thought

5_20

that after many iterations, the system would exhibit thermalization, an ergodic behavior in which

5_21

the influence of the initial modes of vibration fade and the system becomes more or less random

5_22

with all modes excited more or less equally. Instead, the system exhibited a very complicated

5_23

quasi-periodic behavior. They published their results in a Los Alamos technical report in 1955.

5_24

(Enrico Fermi died in 1954, and so this technical report was published after Fermi's death.)

5_25

In 2020, National Security Science magazine featured an article on Tsingou that included her

5_26

commentary and historical reflections on the FPUT problem. In the article,Tsingou states “I

5_27

remember sitting there one day with Pasta and Ulam,” as they brainstormed “some problems we could

5_28

do on the computer, some really mathematical problems.” They tried several things, but, eventually,

5_29

“they came up with this vibrating string.”

5_30

The FPUT experiment was important both in showing the complexity of nonlinear system behavior and

5_31

the value of computer simulation in analyzing systems.

5_32

Name change

5_33

The original paper names Fermi, Pasta, and Ulam as authors (although Fermi died before the report

5_34

was written) with an acknowledgement to Tsingou for her work in programming the MANIAC simulations.

5_35

Mary Tsingou's contributions to the FPUT problem were largely ignored by the community until

5_36

published additional information regarding the development and called for the problem to be renamed

5_37

to grant her attribution as well.

5_38

The FPUT lattice system

5_39

Fermi, Pasta, Ulam, and Tsingou simulated the vibrating string by solving the following discrete

5_40

system of nearest-neighbor coupled oscillators. We follow the explanation as given in Richard

5_41

Palais's article. Let there be N oscillators representing a string of length with equilibrium

5_42

positions , where is the lattice spacing. Then the position of the j-th oscillator as a function

5_43

of time is , so that gives the displacement from equilibrium. FPUT used the following equations of

5_44

motion: